{
 "metadata": {
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.3-final"
  },
  "orig_nbformat": 2,
  "kernelspec": {
   "name": "python3",
   "display_name": "Python 3",
   "language": "python"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2,
 "cells": [
  {
   "source": [
    "<center><h1>第一章 预备知识</h1></center>"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "source": [
    "## 一、Python基础\n",
    "### 1. 列表推导式与条件赋值\n",
    "\n",
    "在生成一个数字序列的时候，在`Python`中可以如下写出："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "[0, 2, 4, 6, 8]"
      ]
     },
     "metadata": {},
     "execution_count": 1
    }
   ],
   "source": [
    "L = []\n",
    "def my_func(x):\n",
    "    return 2*x\n",
    "for i in range(5):\n",
    "    L.append(my_func(i))\n",
    "L"
   ]
  },
  {
   "source": [
    "事实上可以利用列表推导式进行写法上的简化：`[* for i in *]`。其中，第一个`*`为映射函数，其输入为后面`i`指代的内容，第二个`*`表示迭代的对象。"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "[0, 2, 4, 6, 8]"
      ]
     },
     "metadata": {},
     "execution_count": 2
    }
   ],
   "source": [
    "[my_func(i) for i in range(5)]"
   ]
  },
  {
   "source": [
    "列表表达式还支持多层嵌套，如下面的例子中第一个`for`为外层循环，第二个为内层循环："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "['a_c', 'a_d', 'b_c', 'b_d']"
      ]
     },
     "metadata": {},
     "execution_count": 3
    }
   ],
   "source": [
    "[m+'_'+n for m in ['a', 'b'] for n in ['c', 'd']]"
   ]
  },
  {
   "source": [
    "除了列表推导式，另一个实用的语法糖是带有`if`选择的条件赋值，其形式为`value = a if condition else b`："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "'cat'"
      ]
     },
     "metadata": {},
     "execution_count": 4
    }
   ],
   "source": [
    "value = 'cat' if 2>1 else 'dog'\n",
    "value"
   ]
  },
  {
   "source": [
    "等价于如下的写法：\n",
    "```python\n",
    "a, b = 'cat', 'dog'\n",
    "condition = 2 > 1 # 此时为True\n",
    "if condition:\n",
    "    value = a\n",
    "else:\n",
    "    value = b\n",
    "```"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "source": [
    "下面举一个例子，截断列表中超过5的元素，即超过5的用5代替，小于5的保留原来的值："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "[1, 2, 3, 4, 5, 5, 5]"
      ]
     },
     "metadata": {},
     "execution_count": 5
    }
   ],
   "source": [
    "L = [1, 2, 3, 4, 5, 6, 7]\n",
    "[i if i <= 5 else 5 for i in L]"
   ]
  },
  {
   "source": [
    "### 2. 匿名函数与map方法\n",
    "\n",
    "有一些函数的定义具有清晰简单的映射关系，例如上面的`my_func`函数，这时候可以用匿名函数的方法简洁地表示："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "metadata": {},
     "execution_count": 6
    }
   ],
   "source": [
    "my_func = lambda x: 2*x\n",
    "my_func(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "metadata": {},
     "execution_count": 7
    }
   ],
   "source": [
    "multi_para_func = lambda a, b: a + b\n",
    "multi_para_func(1, 2) "
   ]
  },
  {
   "source": [
    "但上面的用法其实违背了“匿名”的含义，事实上它往往在无需多处调用的场合进行使用，例如上面列表推导式中的例子，用户不关心函数的名字，只关心这种映射的关系："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "[0, 2, 4, 6, 8]"
      ]
     },
     "metadata": {},
     "execution_count": 8
    }
   ],
   "source": [
    "[(lambda x: 2*x)(i) for i in range(5)]"
   ]
  },
  {
   "source": [
    "对于上述的这种列表推导式的匿名函数映射，`Python`中提供了`map`函数来完成，它返回的是一个`map`对象，需要通过`list`转为列表："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "[0, 2, 4, 6, 8]"
      ]
     },
     "metadata": {},
     "execution_count": 9
    }
   ],
   "source": [
    "list(map(lambda x: 2*x, range(5)))"
   ]
  },
  {
   "source": [
    "对于多个输入值的函数映射，可以通过追加迭代对象实现："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "['0_a', '1_b', '2_c', '3_d', '4_e']"
      ]
     },
     "metadata": {},
     "execution_count": 10
    }
   ],
   "source": [
    "list(map(lambda x, y: str(x)+'_'+y, range(5), list('abcde')))"
   ]
  },
  {
   "source": [
    "### 3. zip对象与enumerate方法\n",
    "\n",
    "`zip`函数能够把多个可迭代对象打包成一个元组构成的可迭代对象，它返回了一个`zip`对象，通过`tuple`, `list`可以得到相应的打包结果："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "[('a', 'd', 'h'), ('b', 'e', 'i'), ('c', 'f', 'j')]"
      ]
     },
     "metadata": {},
     "execution_count": 11
    }
   ],
   "source": [
    "L1, L2, L3 = list('abc'), list('def'), list('hij')\n",
    "list(zip(L1, L2, L3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "(('a', 'd', 'h'), ('b', 'e', 'i'), ('c', 'f', 'j'))"
      ]
     },
     "metadata": {},
     "execution_count": 12
    }
   ],
   "source": [
    "tuple(zip(L1, L2, L3))"
   ]
  },
  {
   "source": [
    "往往会在循环迭代的时候使用到`zip`函数："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "a d h\nb e i\nc f j\n"
     ]
    }
   ],
   "source": [
    "for i, j, k in zip(L1, L2, L3):\n",
    "     print(i, j, k)"
   ]
  },
  {
   "source": [
    "`enumerate`是一种特殊的打包，它可以在迭代时绑定迭代元素的遍历序号："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "0 a\n1 b\n2 c\n3 d\n"
     ]
    }
   ],
   "source": [
    "L = list('abcd')\n",
    "for index, value in enumerate(L):\n",
    "     print(index, value)"
   ]
  },
  {
   "source": [
    "用`zip`对象也能够简单地实现这个功能："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "0 a\n1 b\n2 c\n3 d\n"
     ]
    }
   ],
   "source": [
    "for index, value in zip(range(len(L)), L):\n",
    "     print(index, value)"
   ]
  },
  {
   "source": [
    "当需要对两个列表建立字典映射时，可以利用`zip`对象："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "{'a': 'd', 'b': 'e', 'c': 'f'}"
      ]
     },
     "metadata": {},
     "execution_count": 16
    }
   ],
   "source": [
    "dict(zip(L1, L2))"
   ]
  },
  {
   "source": [
    "既然有了压缩函数，那么`Python`也提供了`*`操作符和`zip`联合使用来进行解压操作："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "[('a', 'd', 'h'), ('b', 'e', 'i'), ('c', 'f', 'j')]"
      ]
     },
     "metadata": {},
     "execution_count": 17
    }
   ],
   "source": [
    "zipped = list(zip(L1, L2, L3))\n",
    "zipped"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "[('a', 'b', 'c'), ('d', 'e', 'f'), ('h', 'i', 'j')]"
      ]
     },
     "metadata": {},
     "execution_count": 18
    }
   ],
   "source": [
    "list(zip(*zipped)) # 三个元组分别对应原来的列表"
   ]
  },
  {
   "source": [
    "## 二、Numpy基础\n",
    "### 1. np数组的构造\n",
    "最一般的方法是通过`array`来构造："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([1, 2, 3])"
      ]
     },
     "metadata": {},
     "execution_count": 19
    }
   ],
   "source": [
    "import numpy as np\n",
    "np.array([1,2,3])"
   ]
  },
  {
   "source": [
    "下面讨论一些特殊数组的生成方式：\n",
    "\n",
    "【a】等差序列：`np.linspace`, `np.arange`"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([1. , 1.4, 1.8, 2.2, 2.6, 3. , 3.4, 3.8, 4.2, 4.6, 5. ])"
      ]
     },
     "metadata": {},
     "execution_count": 20
    }
   ],
   "source": [
    "np.linspace(1,5,11) # 起始、终止（包含）、样本个数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([1, 3])"
      ]
     },
     "metadata": {},
     "execution_count": 21
    }
   ],
   "source": [
    "np.arange(1,5,2) # 起始、终止（不包含）、步长"
   ]
  },
  {
   "source": [
    "【b】特殊矩阵：`zeros`, `eye`, `full`"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0., 0., 0.],\n",
       "       [0., 0., 0.]])"
      ]
     },
     "metadata": {},
     "execution_count": 22
    }
   ],
   "source": [
    "np.zeros((2,3)) # 传入元组表示各维度大小"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[1., 0., 0.],\n",
       "       [0., 1., 0.],\n",
       "       [0., 0., 1.]])"
      ]
     },
     "metadata": {},
     "execution_count": 23
    }
   ],
   "source": [
    "np.eye(3) # 3*3的单位矩阵"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0., 1., 0.],\n",
       "       [0., 0., 1.],\n",
       "       [0., 0., 0.]])"
      ]
     },
     "metadata": {},
     "execution_count": 24
    }
   ],
   "source": [
    "np.eye(3, k=1) # 偏移主对角线1个单位的伪单位矩阵"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[10, 10, 10],\n",
       "       [10, 10, 10]])"
      ]
     },
     "metadata": {},
     "execution_count": 25
    }
   ],
   "source": [
    "np.full((2,3), 10) # 元组传入大小，10表示填充数值"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[1, 2, 3],\n",
       "       [1, 2, 3]])"
      ]
     },
     "metadata": {},
     "execution_count": 26
    }
   ],
   "source": [
    "np.full((2,3), [1,2,3]) # 每行填入相同的列表"
   ]
  },
  {
   "source": [
    "【c】随机矩阵：`np.random`\n",
    "\n",
    "最常用的随机生成函数为`rand`, `randn`, `randint`, `choice`，它们分别表示0-1均匀分布的随机数组、标准正态的随机数组、随机整数组和随机列表抽样："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([0.92340835, 0.20019461, 0.40755472])"
      ]
     },
     "metadata": {},
     "execution_count": 27
    }
   ],
   "source": [
    "np.random.rand(3) # 生成服从0-1均匀分布的三个随机数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0.8012362 , 0.53154881, 0.05858554],\n",
       "       [0.13103034, 0.18108091, 0.30253153],\n",
       "       [0.00528884, 0.99402007, 0.36348797]])"
      ]
     },
     "metadata": {},
     "execution_count": 28
    }
   ],
   "source": [
    "np.random.rand(3, 3) # 注意这里传入的不是元组，每个维度大小分开输入"
   ]
  },
  {
   "source": [
    "对于服从区间`a`到`b`上的均匀分布可以如下生成："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([6.59370831, 8.03865138, 9.19172546])"
      ]
     },
     "metadata": {},
     "execution_count": 29
    }
   ],
   "source": [
    "a, b = 5, 15\n",
    "(b - a) * np.random.rand(3) + a"
   ]
  },
  {
   "source": [
    "一般的，可以选择已有的库函数："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([11.26499636, 13.12311185,  6.00774156])"
      ]
     },
     "metadata": {},
     "execution_count": 30
    }
   ],
   "source": [
    "np.random.uniform(5, 15, 3)"
   ]
  },
  {
   "source": [
    "`randn`生成了`N(0,I)`的标准正态分布："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([ 1.87000209,  1.19885561, -0.58802943])"
      ]
     },
     "metadata": {},
     "execution_count": 31
    }
   ],
   "source": [
    "np.random.randn(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[-1.3642839 , -0.31497567],\n",
       "       [-1.9452492 , -3.17272882]])"
      ]
     },
     "metadata": {},
     "execution_count": 32
    }
   ],
   "source": [
    "np.random.randn(2, 2)"
   ]
  },
  {
   "source": [
    "对于服从方差为$\\sigma^2$均值为$\\mu$的一元正态分布可以如下生成："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([1.56024917, 0.22829486, 7.3764211 ])"
      ]
     },
     "metadata": {},
     "execution_count": 33
    }
   ],
   "source": [
    "sigma, mu = 2.5, 3\n",
    "mu + np.random.randn(3) * sigma"
   ]
  },
  {
   "source": [
    "同样的，也可选择从已有函数生成："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([3.53517851, 5.3441269 , 3.51192744])"
      ]
     },
     "metadata": {},
     "execution_count": 34
    }
   ],
   "source": [
    "np.random.normal(3, 2.5, 3)"
   ]
  },
  {
   "source": [
    "`randint`可以指定生成随机整数的最小值最大值（不包含）和维度大小："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[ 5, 12],\n",
       "       [14,  9]])"
      ]
     },
     "metadata": {},
     "execution_count": 35
    }
   ],
   "source": [
    "low, high, size = 5, 15, (2,2) # 生成5到14的随机整数\n",
    "np.random.randint(low, high, size)"
   ]
  },
  {
   "source": [
    "`choice`可以从给定的列表中，以一定概率和方式抽取结果，当不指定概率时为均匀采样，默认抽取方式为有放回抽样："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array(['b', 'a'], dtype='<U1')"
      ]
     },
     "metadata": {},
     "execution_count": 36
    }
   ],
   "source": [
    "my_list = ['a', 'b', 'c', 'd']\n",
    "np.random.choice(my_list, 2, replace=False, p=[0.1, 0.7, 0.1 ,0.1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([['c', 'b', 'd'],\n",
       "       ['d', 'a', 'd'],\n",
       "       ['a', 'c', 'd']], dtype='<U1')"
      ]
     },
     "metadata": {},
     "execution_count": 37
    }
   ],
   "source": [
    "np.random.choice(my_list, (3,3))"
   ]
  },
  {
   "source": [
    "当返回的元素个数与原列表相同时，不放回抽样等价于使用`permutation`函数，即打散原列表："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array(['c', 'a', 'd', 'b'], dtype='<U1')"
      ]
     },
     "metadata": {},
     "execution_count": 38
    }
   ],
   "source": [
    "np.random.permutation(my_list)"
   ]
  },
  {
   "source": [
    "最后，需要提到的是随机种子，它能够固定随机数的输出结果："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "0.5488135039273248"
      ]
     },
     "metadata": {},
     "execution_count": 39
    }
   ],
   "source": [
    "np.random.seed(0)\n",
    "np.random.rand()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "0.5488135039273248"
      ]
     },
     "metadata": {},
     "execution_count": 40
    }
   ],
   "source": [
    "np.random.seed(0)\n",
    "np.random.rand()"
   ]
  },
  {
   "source": [
    "### 2. np数组的变形与合并\n",
    "【a】转置：`T`"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0., 0.],\n",
       "       [0., 0.],\n",
       "       [0., 0.]])"
      ]
     },
     "metadata": {},
     "execution_count": 41
    }
   ],
   "source": [
    "np.zeros((2,3)).T"
   ]
  },
  {
   "source": [
    "【b】合并操作：`r_`, `c_`\n",
    "\n",
    "对于二维数组而言，`r_`和`c_`分别表示上下合并和左右合并："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0., 0., 0.],\n",
       "       [0., 0., 0.],\n",
       "       [0., 0., 0.],\n",
       "       [0., 0., 0.]])"
      ]
     },
     "metadata": {},
     "execution_count": 42
    }
   ],
   "source": [
    "np.r_[np.zeros((2,3)),np.zeros((2,3))]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0., 0., 0., 0., 0., 0.],\n",
       "       [0., 0., 0., 0., 0., 0.]])"
      ]
     },
     "metadata": {},
     "execution_count": 43
    }
   ],
   "source": [
    "np.c_[np.zeros((2,3)),np.zeros((2,3))]"
   ]
  },
  {
   "source": [
    "一维数组和二维数组进行合并时，应当把其视作列向量，在长度匹配的情况下只能够使用左右合并的`c_`操作："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "ValueError('all the input arrays must have same number of dimensions, but the array at index 0 has 1 dimension(s) and the array at index 1 has 2 dimension(s)')"
      ]
     },
     "metadata": {},
     "execution_count": 44
    }
   ],
   "source": [
    "try:\n",
    "     np.r_[np.array([0,0]),np.zeros((2,1))]\n",
    "except Exception as e:\n",
    "     Err_Msg = e\n",
    "Err_Msg"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([0., 0., 0., 0.])"
      ]
     },
     "metadata": {},
     "execution_count": 45
    }
   ],
   "source": [
    "np.r_[np.array([0,0]),np.zeros(2)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0., 0., 0., 0.],\n",
       "       [0., 0., 0., 0.]])"
      ]
     },
     "metadata": {},
     "execution_count": 46
    }
   ],
   "source": [
    "np.c_[np.array([0,0]),np.zeros((2,3))]"
   ]
  },
  {
   "source": [
    "【c】维度变换：`reshape`\n",
    "\n",
    "`reshape`能够帮助用户把原数组按照新的维度重新排列。在使用时有两种模式，分别为`C`模式和`F`模式，分别以逐行和逐列的顺序进行填充读取。"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0, 1, 2, 3],\n",
       "       [4, 5, 6, 7]])"
      ]
     },
     "metadata": {},
     "execution_count": 47
    }
   ],
   "source": [
    "target = np.arange(8).reshape(2,4)\n",
    "target"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0, 1],\n",
       "       [2, 3],\n",
       "       [4, 5],\n",
       "       [6, 7]])"
      ]
     },
     "metadata": {},
     "execution_count": 48
    }
   ],
   "source": [
    "target.reshape((4,2), order='C') # 按照行读取和填充"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0, 2],\n",
       "       [4, 6],\n",
       "       [1, 3],\n",
       "       [5, 7]])"
      ]
     },
     "metadata": {},
     "execution_count": 49
    }
   ],
   "source": [
    "target.reshape((4,2), order='F') # 按照列读取和填充"
   ]
  },
  {
   "source": [
    "特别地，由于被调用数组的大小是确定的，`reshape`允许有一个维度存在空缺，此时只需填充-1即可："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0, 1],\n",
       "       [2, 3],\n",
       "       [4, 5],\n",
       "       [6, 7]])"
      ]
     },
     "metadata": {},
     "execution_count": 50
    }
   ],
   "source": [
    "target.reshape((4,-1))"
   ]
  },
  {
   "source": [
    "下面将`n*1`大小的数组转为1维数组的操作是经常使用的："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[1.],\n",
       "       [1.],\n",
       "       [1.]])"
      ]
     },
     "metadata": {},
     "execution_count": 51
    }
   ],
   "source": [
    "target = np.ones((3,1))\n",
    "target"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([1., 1., 1.])"
      ]
     },
     "metadata": {},
     "execution_count": 52
    }
   ],
   "source": [
    "target.reshape(-1)"
   ]
  },
  {
   "source": [
    "### 3. np数组的切片与索引\n",
    "数组的切片模式支持使用`slice`类型的`start:end:step`切片，还可以直接传入列表指定某个维度的索引进行切片："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0, 1, 2],\n",
       "       [3, 4, 5],\n",
       "       [6, 7, 8]])"
      ]
     },
     "metadata": {},
     "execution_count": 53
    }
   ],
   "source": [
    "target = np.arange(9).reshape(3,3)\n",
    "target"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0, 2],\n",
       "       [3, 5]])"
      ]
     },
     "metadata": {},
     "execution_count": 54
    }
   ],
   "source": [
    "target[:-1, [0,2]]"
   ]
  },
  {
   "source": [
    "此外，还可以利用`np.ix_`在对应的维度上使用布尔索引，但此时不能使用`slice`切片："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0, 2],\n",
       "       [6, 8]])"
      ]
     },
     "metadata": {},
     "execution_count": 55
    }
   ],
   "source": [
    "target[np.ix_([True, False, True], [True, False, True])]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[3, 5],\n",
       "       [6, 8]])"
      ]
     },
     "metadata": {},
     "execution_count": 56
    }
   ],
   "source": [
    "target[np.ix_([1,2], [True, False, True])]"
   ]
  },
  {
   "source": [
    "当数组维度为1维时，可以直接进行布尔索引，而无需`np.ix_`："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([0, 2, 4, 6, 8])"
      ]
     },
     "metadata": {},
     "execution_count": 57
    }
   ],
   "source": [
    "new = target.reshape(-1)\n",
    "new[new%2==0]"
   ]
  },
  {
   "source": [
    "### 4. 常用函数\n",
    "为了简单起见，这里假设下述函数输入的数组都是一维的。\n",
    "\n",
    "【a】`where`\n",
    "\n",
    "`where`是一种条件函数，可以指定满足条件与不满足条件位置对应的填充值："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([5, 1, 5, 5])"
      ]
     },
     "metadata": {},
     "execution_count": 58
    }
   ],
   "source": [
    "a = np.array([-1,1,-1,0])\n",
    "np.where(a>0, a, 5) # 对应位置为True时填充a对应元素，否则填充5"
   ]
  },
  {
   "source": [
    "【b】`nonzero`, `argmax`, `argmin`\n",
    "\n",
    "这三个函数返回的都是索引，`nonzero`返回非零数的索引，`argmax`, `argmin`分别返回最大和最小数的索引："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "(array([0, 1, 3, 4, 5], dtype=int64),)"
      ]
     },
     "metadata": {},
     "execution_count": 59
    }
   ],
   "source": [
    "a = np.array([-2,-5,0,1,3,-1])\n",
    "np.nonzero(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "metadata": {},
     "execution_count": 60
    }
   ],
   "source": [
    "a.argmax()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "metadata": {},
     "execution_count": 61
    }
   ],
   "source": [
    "a.argmin()"
   ]
  },
  {
   "source": [
    "【c】`any`, `all`\n",
    "\n",
    "`any`指当序列至少 **存在一个** `True`或非零元素时返回`True`，否则返回`False`\n",
    "\n",
    "`all`指当序列元素 **全为** `True`或非零元素时返回`True`，否则返回`False`"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "metadata": {},
     "execution_count": 62
    }
   ],
   "source": [
    "a = np.array([0,1])\n",
    "a.any()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "metadata": {},
     "execution_count": 63
    }
   ],
   "source": [
    " a.all()"
   ]
  },
  {
   "source": [
    "【d】`cumprod`, `cumsum`, `diff`\n",
    "\n",
    "`cumprod`, `cumsum`分别表示累乘和累加函数，返回同长度的数组，`diff`表示和前一个元素做差，由于第一个元素为缺失值，因此在默认参数情况下，返回长度是原数组减1"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([1, 2, 6], dtype=int32)"
      ]
     },
     "metadata": {},
     "execution_count": 64
    }
   ],
   "source": [
    "a = np.array([1,2,3])\n",
    "a.cumprod()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([1, 3, 6], dtype=int32)"
      ]
     },
     "metadata": {},
     "execution_count": 65
    }
   ],
   "source": [
    "a.cumsum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([1, 1])"
      ]
     },
     "metadata": {},
     "execution_count": 66
    }
   ],
   "source": [
    "np.diff(a)"
   ]
  },
  {
   "source": [
    "【e】 统计函数\n",
    "\n",
    "常用的统计函数包括`max, min, mean, median, std, var, sum, quantile`，其中分位数计算是全局方法，因此不能通过`array.quantile`的方法调用："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([0, 1, 2, 3, 4])"
      ]
     },
     "metadata": {},
     "execution_count": 67
    }
   ],
   "source": [
    "target = np.arange(5)\n",
    "target"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "metadata": {},
     "execution_count": 68
    }
   ],
   "source": [
    "target.max()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "2.0"
      ]
     },
     "metadata": {},
     "execution_count": 69
    }
   ],
   "source": [
    "np.quantile(target, 0.5) # 0.5分位数"
   ]
  },
  {
   "source": [
    "但是对于含有缺失值的数组，它们返回的结果也是缺失值，如果需要略过缺失值，必须使用`nan*`类型的函数，上述的几个统计函数都有对应的`nan*`函数。"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([ 1.,  2., nan])"
      ]
     },
     "metadata": {},
     "execution_count": 70
    }
   ],
   "source": [
    "target = np.array([1, 2, np.nan])\n",
    "target"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "nan"
      ]
     },
     "metadata": {},
     "execution_count": 71
    }
   ],
   "source": [
    "target.max()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "2.0"
      ]
     },
     "metadata": {},
     "execution_count": 72
    }
   ],
   "source": [
    "np.nanmax(target)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "1.5"
      ]
     },
     "metadata": {},
     "execution_count": 73
    }
   ],
   "source": [
    "np.nanquantile(target, 0.5)"
   ]
  },
  {
   "source": [
    "对于协方差和相关系数分别可以利用`cov, corrcoef`如下计算："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[ 11.66666667, -16.66666667],\n",
       "       [-16.66666667,  38.66666667]])"
      ]
     },
     "metadata": {},
     "execution_count": 74
    }
   ],
   "source": [
    "target1 = np.array([1,3,5,9])\n",
    "target2 = np.array([1,5,3,-9])\n",
    "np.cov(target1, target2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[ 1.        , -0.78470603],\n",
       "       [-0.78470603,  1.        ]])"
      ]
     },
     "metadata": {},
     "execution_count": 75
    }
   ],
   "source": [
    "np.corrcoef(target1, target2)"
   ]
  },
  {
   "source": [
    "最后，需要说明二维`Numpy`数组中统计函数的`axis`参数，它能够进行某一个维度下的统计特征计算，当`axis=0`时结果为列的统计指标，当`axis=1`时结果为行的统计指标："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[1, 2, 3],\n",
       "       [4, 5, 6],\n",
       "       [7, 8, 9]])"
      ]
     },
     "metadata": {},
     "execution_count": 76
    }
   ],
   "source": [
    "target = np.arange(1,10).reshape(3,-1)\n",
    "target"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([12, 15, 18])"
      ]
     },
     "metadata": {},
     "execution_count": 77
    }
   ],
   "source": [
    "target.sum(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([ 6, 15, 24])"
      ]
     },
     "metadata": {},
     "execution_count": 78
    }
   ],
   "source": [
    "target.sum(1)"
   ]
  },
  {
   "source": [
    "### 5. 广播机制\n",
    "\n",
    "广播机制用于处理两个不同维度数组之间的操作，这里只讨论不超过两维的数组广播机制。\n",
    "\n",
    "【a】标量和数组的操作\n",
    "\n",
    "当一个标量和数组进行运算时，标量会自动把大小扩充为数组大小，之后进行逐元素操作："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[4., 4.],\n",
       "       [4., 4.]])"
      ]
     },
     "metadata": {},
     "execution_count": 79
    }
   ],
   "source": [
    "res = 3 * np.ones((2,2)) + 1\n",
    "res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0.25, 0.25],\n",
       "       [0.25, 0.25]])"
      ]
     },
     "metadata": {},
     "execution_count": 80
    }
   ],
   "source": [
    "res = 1 / res\n",
    "res"
   ]
  },
  {
   "source": [
    "【b】二维数组之间的操作\n",
    "\n",
    "当两个数组维度完全一致时，使用对应元素的操作，否则会报错，除非其中的某个数组的维度是$m×1$或者$1×n$，那么会扩充其具有$1$的维度为另一个数组对应维度的大小。例如，$1×2$数组和$3×2$数组做逐元素运算时会把第一个数组扩充为$3×2$，扩充时的对应数值进行赋值。但是，需要注意的是，如果第一个数组的维度是$1×3$，那么由于在第二维上的大小不匹配且不为$1$，此时报错。"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[1., 1.],\n",
       "       [1., 1.],\n",
       "       [1., 1.]])"
      ]
     },
     "metadata": {},
     "execution_count": 81
    }
   ],
   "source": [
    "res = np.ones((3,2))\n",
    "res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[2., 3.],\n",
       "       [2., 3.],\n",
       "       [2., 3.]])"
      ]
     },
     "metadata": {},
     "execution_count": 82
    }
   ],
   "source": [
    "res * np.array([[2,3]]) # 第二个数组扩充第一维度为3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[2., 2.],\n",
       "       [3., 3.],\n",
       "       [4., 4.]])"
      ]
     },
     "metadata": {},
     "execution_count": 83
    }
   ],
   "source": [
    "res * np.array([[2],[3],[4]]) # 第二个数组扩充第二维度为2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[2., 2.],\n",
       "       [2., 2.],\n",
       "       [2., 2.]])"
      ]
     },
     "metadata": {},
     "execution_count": 84
    }
   ],
   "source": [
    "res * np.array([[2]]) # 等价于两次扩充，第二个数组两个维度分别扩充为3和2"
   ]
  },
  {
   "source": [
    "【c】一维数组与二维数组的操作\n",
    "\n",
    "当一维数组$A_k$与二维数组$B_{m,n}$操作时，等价于把一维数组视作$A_{1,k}$的二维数组，使用的广播法则与【b】中一致，当$k!=n$且$k,n$都不是$1$时报错。"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[2., 2., 2.],\n",
       "       [2., 2., 2.]])"
      ]
     },
     "metadata": {},
     "execution_count": 85
    }
   ],
   "source": [
    "np.ones(3) + np.ones((2,3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[2., 2., 2.],\n",
       "       [2., 2., 2.]])"
      ]
     },
     "metadata": {},
     "execution_count": 86
    }
   ],
   "source": [
    "np.ones(3) + np.ones((2,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[2., 2., 2.],\n",
       "       [2., 2., 2.]])"
      ]
     },
     "metadata": {},
     "execution_count": 87
    }
   ],
   "source": [
    "np.ones(1) + np.ones((2,3))"
   ]
  },
  {
   "source": [
    "### 6. 向量与矩阵的计算\n",
    "【a】向量内积：`dot`\n",
    "\n",
    "$$\\rm \\mathbf{a}\\cdot\\mathbf{b} = \\sum_ia_ib_i$$"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "22"
      ]
     },
     "metadata": {},
     "execution_count": 88
    }
   ],
   "source": [
    "a = np.array([1,2,3])\n",
    "b = np.array([1,3,5])\n",
    "a.dot(b)"
   ]
  },
  {
   "source": [
    "【b】向量范数和矩阵范数：`np.linalg.norm`\n",
    "\n",
    "在矩阵范数的计算中，最重要的是`ord`参数，可选值如下："
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "source": [
    "| ord | norm for matrices | norm for vectors |\n",
    "| :---- | ----: | ----: |\n",
    "| None   | Frobenius norm | 2-norm |\n",
    "| 'fro'  | Frobenius norm  | / |\n",
    "| 'nuc'  | nuclear norm    | / |\n",
    "| inf    | max(sum(abs(x), axis=1))   | max(abs(x)) |\n",
    "| -inf   | min(sum(abs(x), axis=1))  |  min(abs(x)) |\n",
    "| 0      | /   |  sum(x != 0) |\n",
    "| 1      | max(sum(abs(x), axis=0))  |  as below |\n",
    "| -1     | min(sum(abs(x), axis=0))   |  as below |\n",
    "| 2      | 2-norm (largest sing. value) | as below |\n",
    "| -2     | smallest singular value    | as below |\n",
    "| other  | /   | sum(abs(x)\\*\\*ord)\\*\\*(1./ord) |"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0, 1],\n",
       "       [2, 3]])"
      ]
     },
     "metadata": {},
     "execution_count": 89
    }
   ],
   "source": [
    "matrix_target =  np.arange(4).reshape(-1,2)\n",
    "matrix_target"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "3.7416573867739413"
      ]
     },
     "metadata": {},
     "execution_count": 90
    }
   ],
   "source": [
    "np.linalg.norm(matrix_target, 'fro')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "5.0"
      ]
     },
     "metadata": {},
     "execution_count": 91
    }
   ],
   "source": [
    "np.linalg.norm(matrix_target, np.inf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "3.702459173643833"
      ]
     },
     "metadata": {},
     "execution_count": 92
    }
   ],
   "source": [
    "np.linalg.norm(matrix_target, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([0, 1, 2, 3])"
      ]
     },
     "metadata": {},
     "execution_count": 93
    }
   ],
   "source": [
    "vector_target =  np.arange(4)\n",
    "vector_target"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "3.0"
      ]
     },
     "metadata": {},
     "execution_count": 94
    }
   ],
   "source": [
    "np.linalg.norm(vector_target, np.inf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "3.7416573867739413"
      ]
     },
     "metadata": {},
     "execution_count": 95
    }
   ],
   "source": [
    "np.linalg.norm(vector_target, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "3.3019272488946263"
      ]
     },
     "metadata": {},
     "execution_count": 96
    }
   ],
   "source": [
    "np.linalg.norm(vector_target, 3)"
   ]
  },
  {
   "source": [
    "【c】矩阵乘法：`@`\n",
    "\n",
    "$$\\rm [\\mathbf{A}_{m\\times p}\\mathbf{B}_{p\\times n}]_{ij} = \\sum_{k=1}^p\\mathbf{A}_{ik}\\mathbf{B}_{kj}$$"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[0, 1],\n",
       "       [2, 3]])"
      ]
     },
     "metadata": {},
     "execution_count": 97
    }
   ],
   "source": [
    "a = np.arange(4).reshape(-1,2)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[-4, -3],\n",
       "       [-2, -1]])"
      ]
     },
     "metadata": {},
     "execution_count": 98
    }
   ],
   "source": [
    "b = np.arange(-4,0).reshape(-1,2)\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "array([[ -2,  -1],\n",
       "       [-14,  -9]])"
      ]
     },
     "metadata": {},
     "execution_count": 99
    }
   ],
   "source": [
    "a@b"
   ]
  },
  {
   "source": [
    "## 三、练习\n",
    "### Ex1：利用列表推导式写矩阵乘法\n",
    "一般的矩阵乘法根据公式，可以由三重循环写出，请将其改写为列表推导式的形式。"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "metadata": {},
     "execution_count": 100
    }
   ],
   "source": [
    "M1 = np.random.rand(2,3)\n",
    "M2 = np.random.rand(3,4)\n",
    "res = np.empty((M1.shape[0],M2.shape[1]))\n",
    "for i in range(M1.shape[0]):\n",
    "    for j in range(M2.shape[1]):\n",
    "        item = 0\n",
    "        for k in range(M1.shape[1]):\n",
    "            item += M1[i][k] * M2[k][j]\n",
    "        res[i][j] = item\n",
    "(np.abs((M1@M2 - res) < 1e-15)).all() # 排除数值误差"
   ]
  },
  {
   "source": [
    "### Ex2：更新矩阵\n",
    "设矩阵 $A_{m×n}$ ，现在对 $A$ 中的每一个元素进行更新生成矩阵 $B$ ，更新方法是 $B_{ij}=A_{ij}\\sum_{k=1}^n\\frac{1}{A_{ik}}$ ，例如下面的矩阵为 $A$ ，则 $B_{2,2}=5\\times(\\frac{1}{4}+\\frac{1}{5}+\\frac{1}{6})=\\frac{37}{12}$ ，请利用 `Numpy` 高效实现。\n",
    "$$\\begin{split}A=\\left[ \\begin{matrix} 1 & 2 &3\\\\4&5&6\\\\7&8&9 \\end{matrix} \\right]\\end{split}$$"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "source": [
    "### Ex3：卡方统计量\n",
    "\n",
    "设矩阵$A_{m\\times n}$，记$B_{ij} = \\frac{(\\sum_{i=1}^mA_{ij})\\times (\\sum_{j=1}^nA_{ij})}{\\sum_{i=1}^m\\sum_{j=1}^nA_{ij}}$，定义卡方值如下：\n",
    "$$\\chi^2 = \\sum_{i=1}^m\\sum_{j=1}^n\\frac{(A_{ij}-B_{ij})^2}{B_{ij}}$$\n",
    "请利用`Numpy`对给定的矩阵$A$计算$\\chi^2$ "
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(0)\n",
    "A = np.random.randint(10, 20, (8, 5))"
   ]
  },
  {
   "source": [
    "### Ex4：改进矩阵计算的性能\n",
    "设$Z$为$m×n$的矩阵，$B$和$U$分别是$m×p$和$p×n$的矩阵，$B_i$为$B$的第$i$行，$U_j$为$U$的第$j$列，下面定义$\\displaystyle R=\\sum_{i=1}^m\\sum_{j=1}^n\\|B_i-U_j\\|_2^2Z_{ij}$，其中$\\|\\mathbf{a}\\|_2^2$表示向量$a$的分量平方和$\\sum_i a_i^2$。\n",
    "\n",
    "现有某人根据如下给定的样例数据计算$R$的值，请充分利用`Numpy`中的函数，基于此问题改进这段代码的性能。"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "100566"
      ]
     },
     "metadata": {},
     "execution_count": 102
    }
   ],
   "source": [
    "np.random.seed(0)\n",
    "m, n, p = 100, 80, 50\n",
    "B = np.random.randint(0, 2, (m, p))\n",
    "U = np.random.randint(0, 2, (p, n))\n",
    "Z = np.random.randint(0, 2, (m, n))\n",
    "def solution(B=B, U=U, Z=Z):\n",
    "    L_res = []\n",
    "    for i in range(m):\n",
    "        for j in range(n):\n",
    "            norm_value = ((B[i]-U[:,j])**2).sum()\n",
    "            L_res.append(norm_value*Z[i][j])\n",
    "    return sum(L_res)\n",
    "solution(B, U, Z)"
   ]
  },
  {
   "source": [
    "### Ex5：连续整数的最大长度\n",
    "\n",
    "输入一个整数的`Numpy`数组，返回其中严格递增连续整数子数组的最大长度，正向是指递增方向。例如，输入\\[1,2,5,6,7\\]，\\[5,6,7\\]为具有最大长度的连续整数子数组，因此输出3；输入\\[3,2,1,2,3,4,6\\]，\\[1,2,3,4\\]为具有最大长度的连续整数子数组，因此输出4。请充分利用`Numpy`的内置函数完成。（提示：考虑使用`nonzero, diff`函数）"
   ],
   "cell_type": "markdown",
   "metadata": {}
  }
 ]
}